I don't understand the use of the French preposition à with articles. Sometimes the article and the preposition stay separate, as in Elle reste à la maison, and sometimes they merge, as in Je vais en France. In other cases, the article is altogether absent, as in Je suis à bout de souffle and à court terme. Could someone logically explain to me how I can differentiate between these cases.

1 Answer
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There is no merging process involved; if you mean by merging something as in "à le => au", that is called rather "contraction" and that has nothing to do with whether to use "à la" or "en"; "à" is one preposition used among other things to indicate the place where something is and also the place where someone/something goes and also to give indications of time. "En" is another preposition, and that one is used to indicate basically a place where someone/something is or into which someone/something goes and also to give indications of time. The meaning of this one is comparable to that of "dans".

A/à la, à l'

For nouns that are not the nouns of countries or regions which gender is feminine (Amérique, Allemagne, Belgique,…) one of the two forms is used according as the noun begins with a consonant or not; if the consonant is an h, then you use "à l'" when the h is a so called "h muet".